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Mathematics & Economics

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Lab Report

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# Analysis Paper: Lab Report and Discussion on the Results (Lab Report Sample)

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Lab report and discussion on the results

source..Content:

Graphical analysis

Studentâ€™s name

Institutional Affiliation

GRAPHING AND GRAPHICAL ANALYSIS

Objectives

The major objectives of this experiment are:

1 To understand graphs and graphical relationships.

2 To understand the use of error bars on graphs.

3 To recognize linear relationships, power relationships, and exponential relationships from the shape of the respective curves and to plot and determine the equations of such curves.

Introduction and Theory

Many experiments are of the type where one wishes to discover the relationship between a pair of quantities that depend on each other, either to verify a theoretical prediction or to help guide the development of a theory where none exists. Examples of related pairs of quantities are: the velocity vs. time of a mass falling in a gravitational field, the period of oscillation vs. mass for a mass-spring system undergoing simple harmonic motion, and the specific heat of a substance vs. its temperature (Tukey & J. 1977).

1 Functional Relationship

There are many kinds of possible relationshipsy=f(x), but we will be concerned here with only three types:

* Linear Relationship

A relationship is linear when the independent variable,x, is related to dependent variable,y, in the form y=ax+b

Where a and b are constants.

* Power Relationship

Here the independent variable is related to the dependent variable as

y=bxa

* Exponential Relationships

Here the independent variable is related to the dependent variable as

y=beax

In order to determine the appropriate relationship, one can refer to theory when possible, or construct a graph and proceed by trial and error. In the second case, one simply graphs y vs. x in excel. If the two quantities are related linearly, a straight line is obtained; otherwise, a curved line results, its general shape and position with respect to the axes can point to the appropriate relationship.

II. Error Bars-part I

Error bars are graphical representations of the uncertainties in the data you are plotting. They facilitate the choice of which particular type of function best describes the relationship between the variables you are plotting. This is particularly true for linear functions. In order for the function to describe your data relationship correctly the curve must pass through most of the error bars (Peakall et al, 2006).

1 Errors Bars-part II

If Î´x is the uncertainty inx, thenÎ´f, the uncertainty inf(x), is given by

Î´f=fxÂ±Î´x-f(x).

Laboratory report

Data Series A - Amplitude vs. Time

Steep/Shallow data

t (s)

Â±

A (cm)

Â±

Asteep

Ashallow

0.00

0.01

50

3%

51.5

48.5

1.00

0.01

45

3%

46.35

43.65

2.00

0.01

40

3%

41.2

38.8

3.00

0.01

36

3%

37.08

34.92

4.00

0.01

33

3%

33.99

32.01

5.00

0.01

30

3%

29.1

30.9

6.00

0.01

27

3%

26.19

27.81

7.00

0.01

25

3%

24.25

25.75

8.00

0.01

23

3%

22.31

23.69

9.00

0.01

21

3%

20.37

21.63

Data Series C - Force vs. Distance

Steep/Shallow data

r (cm)

Â±

F (N)

Â±

Fsteep

Fshallow

2.5

0.1

4.7

0.2

4.9

4.5

3.2

0.1

3.1

0.2

3.3

2.9

3.7

0.1

2.1

0.2

2.3

1.9

5.0

0.1

1.2

0.2

1.4

1

5.2

0.1

1.1

0.2

0.9

1.3

6.1

0.1

0.8

0.2

0.64

1.04

6.5

0.1

0.7

0.2

0.53

0.93

7.4

0.1

0.6

0.2

0.38

0.78

Data Series B - Velocity vs. Position

Steep/Shallow data

x (cm)

Â±

v (m/s)

Â±

vsteep

vshallow

234

8

38

8

46

30

329

8

80

8

88

72

424

8

112

8

120

104

592

8

177

8

177

177

712

8

216

8

208

224

811

8

259

8

251

267

944

8

303

8

295

311

Data Series D - Acceleration vs. Time

Steep/Shallow data

t (ms)

Â±

a (m/s2)

Â±

asteep

ashallow

12

1

15

5

20

10

21

1

22

5

27

17

29

1

30

5

35

25

43

1

49

5

49

49

56

1

86

5

81

91

69

1

141

5

136

146

87

1

269

5

264

274

Summary of results

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